Regularization methods: Our SIDA-RKHS regularization is a type of Tikhonov/ridge regularization that adds a penalty term to the loss function. It differs from previous methods at the penalty term. The commonly used penalty terms include the Euclidean norm in the classical Tikhonov regularization [1]}, [2]}, the RKHS norm with an ad hoc reproducing kernel (often the Gaussian kernel) [3]}, [4]}, the total variation norm in the Rudin-Osher-Fatemi method [5]}, or the \(L^1\)  norm in LASSO [6]}. Whereas each of these penalty terms has their specific applications, none of them take into account of the FSOI, which is fundamental for learning kernels in operators. Also, our regularization method is inspired by the kernel flow method that learns hyper-parameters of the reproducing kernel [7]}, [8]}, [9]}, but our reproducing kernel is determined by the system and the data. Given the importance of regularization to overcome ill-posedness and overfitting, we expect our SIDA-RKHS regularization method to be applicable to a wide range of linear inverse problems and machine learning methods.
